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Useful fact: the restriction of a convex function to a line is a convex function.

Formally, for any point $a$ and any nonzero vector $w$ the function $\varphi(t) = f(a+tw)$ is convex on $\mathbb R$.

If $f$ is $C^1$, then so is $\varphi$. We have $\varphi(t)-\varphi(0)=\varphi'(\xi)\,t>\varphi'(0)\,t$ by the Mean Value Theorem and because $\varphi'$ is increasing. In terms of $f$ this translates into the inequality you want.

If $f$ is $C^2$, then so is $\varphi$. Consequently, $\varphi''(0)\ge 0$ which translates into the inequality you want.